Question

Without actually solving the simultaneous equations given below, decide which simultaneous equations have unique solutions, no solution or infinitely many solutions.
$\left(\mathrm{iv}\right)8y=x-10;2x=3y+7$

Answer 1

$$\begin{gathered} 8y=x-10 \\ 2x=3y+7 \end{gathered}$$

These equations can be written as follows:

$$\begin{gathered} x-8y=10 \\ 2x-3y=7 \end{gathered}$$

Comparing the above equations with the equations $a_{1}x+b_{1}y=c_1\mathrm{and}{a}_{2}x+b_{2}y=c_2,\mathrm{we}\mathrm{get}$:

$$\begin{gathered} a_1=1b_1=-8c_1=10 \\ {a}_2=2b_2=-3c_2=7 \\ \therefore \frac{a_1}{a_2}=\frac{1}{2};\frac{b_1}{b_2}=\frac{-8}{-3}=\frac{8}{3}\mathrm{and}\frac{c_1}{c_2}=\frac{10}{7} \end{gathered}$$

$$\begin{gathered} \mathrm{i}.\mathrm{e}.,\frac{a_1}{a_2}\ne \frac{b_1}{b_2} \\ \therefore \mathrm{The}\mathrm{given}\mathrm{equations}\mathrm{have}\mathrm{unique}\mathrm{solutions}. \end{gathered}$$